High precision phase measurements with optical interferometers play a significant role in both classical measurements[1]}, [2]} and quantum measurements[3]}, [4]}, [5]}, [6]}. Numerous studies show that the quantum measurements with weak value amplification(WVA) can beat the classical schemes with high sensitivity and good robustness[7]}, [8]}, [9]}, [10]}. The concept of weak value in weak measurement was first derived from the work on the arrow of time in quantum theory by Aharonov, Albert and Vaidman in 1988[11]}. In particular, the weak value \(\rm A_w\)  derived by a weak measurement on pre-selected and post-selected can be arbitrarily large, where the coupling strength between the system and the meter is sufficiently small[12]}. Thus, the famous example of the weak measurement of a spin particle leading to a value of 100[11]} has opened up a gate for the WVA in the estimation of small parameters, such as ultra-sensitive beam deflection measurement[7]}, velocity measurements[3]}, ultra-small polarization rotation[16]}, rotation velocity[10]}, single-photon nonlinearity[18]}, precision thermometry[19]}.
